Find two ways to calculate tg15 .

We'll consider 15 degrees as the result of the difference between 2 well known angles: 45 and 30 degrees.

tg 15 = tg(45 - 30) = (tg 45 - tg 30)/[1 + (tg 45*tg30)]

tg 15 = {1 - [(3)^1/2]/3}/[1 + (1*(3)^1/2/3)]

tg 15 = [3 - (3)^1/2]/[3 + (3)^1/2]

We'll multiply the ratio with the adjoint of the denominator, which is [3 - (3)^1/2] and the result will be:

tg 15 = [3 - (3)^1/2]^2/9 - 3

We'll develop the binomial at the numerator:

tg 15 = [9 + 3 - 6*(3)^1/2]/6

tg 15 = [12 - 6*(3)^1/2]/6

tg 15 = 2 - sq root 3

Second method:

We could calculate tg 15 as being the half of 30 degrees.

tg 15 = tg (30/2) = sin (30/2)/cos (30/2)

sin (30/2) = sq root [(1 - cos 30)/2] = sq root{[2 - sq root(3)]/4}

cos (30/2) = sq root [(1 + cos 30)/2] = sq root{[2 + sq root(3)]/4}

tg 15 = {sq root[2 - sq root(3)]}/{sq root[2 + sq root(3)]}

tg 15 = 1/2 + sq root(3)

We'll multiply the denominator with it's adjoint 2 - sq root(3)

tg 15 = 2 - sq root(3)/(4 - 3)

tg 15 = 2 - sq root 3